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Area of a parallelogram or triangle whose sides or diagonals are combinations of a and b

This type of question, asked 6 times: 9 May 2023 S1, 10 May 2023 S1, 11 May 2024 S2, 22 Apr 2025 S1, 15 Apr 2026 S2, 13 May 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    9 May 2023 S1

    142
    If the area of the parallelogram with a⃗\vec{a} and b⃗\vec{b} as two adjacent sides is 15 square units, then the area (in square units) of the parallelogram, having 3a⃗+2b⃗3\vec{a}+2\vec{b} and a⃗+3b⃗\vec{a}+3\vec{b} as two adjacent sides, is

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  2. Question 2

    10 May 2023 S1

    135
    If ∣a⃗∣=2,∣b⃗∣=3|\vec{a}|=2,|\vec{b}|=3 and a⃗,b⃗\vec{a},\vec{b} are mutually perpendicular vectors, then the area of the triangle whose vertices are 0⃗, a⃗+2b⃗, a⃗−2b⃗\vec{0},\,\vec{a}+2\vec{b},\,\vec{a}-2\vec{b} is

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  3. Question 3

    11 May 2024 S2

    118
    If the area of the parallelogram with a⃗\vec{a} and b⃗\vec{b} as two adjacent sides is 16 sq.units, then the area of the parallelogram having 3a⃗+2b⃗3\vec{a}+2\vec{b} and a⃗+3b⃗\vec{a}+3\vec{b} as two adjacent sides (in sq.units) is

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  4. Question 4

    22 Apr 2025 S1

    133
    The area of a parallelogram whose diagonals are the vectors 2aˉ−bˉ2\bar{a}-\bar{b} and 4aˉ−5bˉ4\bar{a}- 5\bar{b}, where aˉ\bar{a} and bˉ\bar{b} are unit vectors forming an angle of 45∘45^{\circ} is

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  5. Question 5

    15 Apr 2026 S2

    90
    Let ABCD be a quadrilateral with AB‾=a⃗,AD‾=b⃗\overline{AB} = \vec{a}, \overline{AD} = \vec{b} and AC‾=3a⃗+2b⃗\overline{AC} = 3\vec{a} + 2\vec{b}. If its area is α\alpha times the area of the parallelogram with AB, AD as adjacent sides, then the value of α\alpha is equal to

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  6. Question 6

    13 May 2026 S1

    88
    Let u⃗\vec{u} and v⃗\vec{v} be unit vectors such that the angle between them is θ=π6\theta = \dfrac{\pi}{6}. If the adjacent sides of a parallelogram are represented by the vectors a⃗=2u⃗+3v⃗\vec{a} = 2\vec{u} + 3\vec{v} and b⃗=u⃗−v⃗\vec{b} = \vec{u} - \vec{v}, then the area of the parallelogram is

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